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Magazine Chapter 10: Problem 36
Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{n+2^{n}}{n^{2} 2^{n}}\end{equation}
Short Answer
Expert verified
The series converges.
Step by step solution
01
Simplify the General Term
Start by simplifying the general term of the series \(a_n = \frac{n+2^n}{n^2 2^n}\). Separate the terms in the numerator to get \(a_n = \frac{n}{n^2 2^n} + \frac{2^n}{n^2 2^n}\). Simplifying gives \(a_n = \frac{1}{n \cdot 2^n} + \frac{1}{n^2}\).
02
Analyze Each Part of the Series
Look at the two parts: \(b_n = \frac{1}{n \cdot 2^n}\) and \(c_n = \frac{1}{n^2}\).
03
Determine Convergence of \(c_n\)
Recognize \(c_n = \frac{1}{n^2}\) as a p-series with \(p = 2 > 1\). A p-series with \(p > 1\) converges. Therefore, \(\sum \frac{1}{n^2}\) converges.
04
Determine Convergence of \(b_n\)
For \(b_n = \frac{1}{n \cdot 2^n}\), compare with a known series. Use the comparison test, comparing it to \(d_n = \frac{1}{2^n}\), which is a geometric series with ratio \(\frac{1}{2}\). The geometric series \(\sum \frac{1}{2^n}\) converges. Since \(\frac{1}{n \cdot 2^n} \leq \frac{1}{2^n}\), by the comparison test, \(\sum b_n\) converges.
05
Combine Results
Since both \(\sum b_n\) and \(\sum c_n\) converge individually, the overall series \(\sum_{n=1}^{\infty} \left( \frac{1}{n \cdot 2^n} + \frac{1}{n^2} \right)\) is the sum of two convergent series, which means the original series converges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
p-series
A p-series is a type of infinite series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( p \) is a positive constant. The convergence of a p-series depends on the value of \( p \). If \( p > 1 \), the p-series converges; however, if \( p \leq 1 \), it diverges.
For example, in the original exercise, the term \( c_n = \frac{1}{n^2} \) is a p-series with \( p = 2 \). Because \( 2 > 1 \), this series converges. Recognizing a p-series is crucial in determining the behavior of an infinite series and is a fundamental concept in calculus. Identifying p-series allows us to use known results about their convergence to aid in solving more complex series problems.
For example, in the original exercise, the term \( c_n = \frac{1}{n^2} \) is a p-series with \( p = 2 \). Because \( 2 > 1 \), this series converges. Recognizing a p-series is crucial in determining the behavior of an infinite series and is a fundamental concept in calculus. Identifying p-series allows us to use known results about their convergence to aid in solving more complex series problems.
geometric series
A geometric series is a series where each term is a constant multiple of the previous term. It takes the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio.
The convergence of a geometric series is straightforward:
The convergence of a geometric series is straightforward:
- If \( |r| < 1 \), the series converges.
- If \( |r| \geq 1 \), the series diverges.
comparison test
The comparison test is a method used to determine the convergence or divergence of an infinite series by comparing it to another series whose convergence is already known.
For this test, two variations are used:
For this test, two variations are used:
- If \( a_n \leq b_n \) for all \( n \) and \( \sum b_n \) converges, then \( \sum a_n \) also converges.
- If \( a_n \geq b_n \) for all \( n \) and \( \sum b_n \) diverges, then \( \sum a_n \) also diverges.
infinite series
An infinite series is the sum of infinite terms and is expressed in the form \( \sum_{n=1}^{\infty} a_n \). The study of such series involves determining whether they add up to a finite value, which implies convergence, or do not, leading to divergence.
Key points about infinite series include:
Key points about infinite series include:
- A series converges if the sequence of partial sums \( S_n = a_1 + a_2 + \ldots + a_n \) approaches a finite limit as \( n \) approaches infinity.
- If the partial sums do not approach a finite limit, the series diverges.
- Various tests, such as the comparison test, p-series test, and geometric series analysis, are utilized to ascertain convergence or divergence.
